Let, \begin{equation} g(\alpha,\beta) = \begin{cases} \frac{\alpha}{\beta}, & \text{if } \alpha > \beta \\ 0, & \text{if } \alpha \leq \beta \end{cases} \end{equation} I want to find the CDF of $$Z=\max\left\{g(X,Y),g(X,C)\right\}$$ where$X,Y\geq0$ are independent random variables, and $C>0$ is a constant.

I am facing two difficulties:

1. I could get the product of CDFs within $\max$ if two were independent. However, as they have $X$ in common, dependency is created. Therefore, I tried to condition on $X$ first and solve. However, in that case, $\frac{X}{C}$ becomes a constant, which is confusing me.
2. I have more complications in applying the conditions of $X>Y$ and $X>C$ seperately to the analysis.

Can someone help me write the CDF of $Z$ in terms of the PDFs/CDFs of $X$ and $Y$?